Compute ∫ V ( x 2 + y 2 + z 2 ) d x d y d z ,where V is the intersection of the spheres x 2 + 2 + z 2 ≤ 1 and x 2 + y 2 + z 2 ≤ 2 z .

Question

Compute      ∫    V    (      x    2    +      y    2    +      z    2    )  d  x  d  y  d  z  ,where V is the intersection of the spheres       x    2        +    2    +      z    2    ≤  1 and       x    2    +      y    2    +      z    2    ≤  2  z  .

Nicolas Calhoun · Accepted Answer

In spherical coordinates,  x  =  ρ  cos  ⁡  θ  sin  ⁡  ϕ  ,  y  =  ρ  sin  ⁡  θ  sin  ⁡  ϕ  ,  z  =  ρ  cos  ⁡  ϕ      S    1    :      x    2    +      y    2    +      z    2    ≤  1      S    2    :      x    2    +      y    2    +      z    2    ≤  2  zAt their intersection,   ρ  =  2  cos  ⁡  ϕ  =  1    ⟹    ϕ  =            π      3      For      0  ≤  ϕ  ≤  π      /    3 is bound above by the sphere       S    1   and for   π      /    3  ≤  ϕ  ≤  π      /    2 is bound above by the sphere       S    2  So the integral can be written as,            ∫      0              2        π                    ∫      0              π                  /                3                    ∫      0      1              ρ      4        sin    ⁡    ϕ         d    ρ         d    ϕ         d    θ         +       Both integrals are straightforward to compute. In the second one, after integrating wrt   ρ, you can substitute   cos  ⁡  ϕ  =  t and then   sin  ⁡  ϕ     d  ϕ  =  −  d  t

Compute ∫ V </msub> ( x 2 </msup> + y 2

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